Range, Interquartile Range and Measures of Dispersion

Range, interquartile range and other measures of dispersion describe how widely data values are spread. In brief: for 2, 4, 4, 4, 5, 5, 7 and 9, the range is 9 − 2 = 7 and, using medians of the two halves, the interquartile range is 6 − 4 = 2. Different measures focus on different parts of the distribution.

A measure of centre alone cannot show whether observations cluster closely or vary greatly. Dispersion adds that missing information. This guide explains range, quartiles, variance, standard deviation and absolute deviations, with explicit calculation conventions and examples that distinguish a whole population from a sample used to estimate a wider population.

Why do datasets need a measure of dispersion?

Two datasets can have the same mean but very different spreads. The values 4, 5, 6 and the values 0, 5, 10 both have mean 5. The first set stays close to that centre, while the second extends much farther away, so the shared mean does not describe their variability.

A dispersion measure summarises separation among values or distance from a centre. Some use only extreme observations, some use the middle portion and others use every value. The choice affects sensitivity to unusual observations and determines what aspect of the distribution the resulting number represents.

Dispersion values are nonnegative, and zero spread means all observations coincide for the standard measures discussed here. Larger spread does not automatically mean worse data or poorer quality; its practical meaning depends on the quantity being measured and the question being asked.

How do you calculate the range?

The range is maximum minus minimum. For 2, 4, 4, 4, 5, 5, 7, 9, the largest value is 9 and the smallest is 2, so the range is 7. It uses the same unit as the observations, such as centimetres for measured lengths.

The range is quick to calculate and shows the full observed span, but it depends on only two observations. Changing an interior value may leave the range unchanged, while changing a single extreme can alter it substantially. It therefore gives little information about how most observations are distributed within the span.

The range also tends to be influenced by sample size because larger samples have more opportunities to include extreme values. Comparing ranges from very different sample sizes should therefore be interpreted cautiously. A zero range indicates identical observations, but matching nonzero ranges do not imply similar distributions.

How do you find quartiles and the interquartile range?

Quartiles divide ordered data into portions around the lower quarter, median and upper quarter. The first quartile Q₁ marks the lower quartile and the third Q₃ marks the upper quartile. The interquartile range is IQR = Q₃ − Q₁, summarising the spread of the central part of the data.

This article uses the median-of-halves convention: sort the observations, find the overall median, then take the median of the lower half and the median of the upper half. For an odd number of observations, exclude the overall median from both halves. Other software and textbooks may use interpolation or different inclusion rules.

For 2, 4, 4, 4, 5, 5, 7, 9, the lower half is 2, 4, 4, 4, giving Q₁ = 4. The upper half is 5, 5, 7, 9, giving Q₃ = 6. The median is 4.5 and the IQR is 6 − 4 = 2.

How do you calculate variance?

Variance measures average squared deviation from the mean, with a denominator chosen according to the statistical purpose. For an entire population of N values, population variance is σ² = Σ(x − μ)²/N. Here μ is the population mean and the sum includes every squared deviation.

For a sample used to estimate population variance, the conventional sample variance is s² = Σ(x − x̄)²/(n − 1), requiring n greater than 1. The n − 1 denominator compensates for estimating the population mean from the same sample under standard independent-sampling assumptions.

For the eight example values, the mean is 5. Their deviations are −3, −1, −1, −1, 0, 0, 2 and 4, whose squares sum to 32. Population variance is 32/8 = 4; sample variance is 32/7, approximately 4.5714. State which interpretation is being used rather than switching denominators silently.

What is standard deviation and how is it interpreted?

Standard deviation is the nonnegative square root of variance. For the example’s population variance of 4, the population standard deviation is 2. Using the sample variance 32/7 gives sample standard deviation √(32/7), approximately 2.1381.

Squaring deviations prevents positive and negative differences from cancelling, and taking the square root restores the original measurement unit. Variance has squared units, while standard deviation has the same unit as the observations. This makes standard deviation easier to compare with actual distances from the mean.

Standard deviation is a measure of spread around the mean, but it is not generally the average absolute distance from the mean. Nor does it guarantee a particular percentage of values within one standard deviation without assumptions about the distribution. Distribution-specific rules should not be applied automatically to every dataset.

How do absolute deviations measure spread?

The mean absolute deviation from the mean averages |x − x̄| rather than squared deviations. For the example with mean 5, the absolute deviations are 3, 1, 1, 1, 0, 0, 2 and 4. Their sum is 12, so the mean absolute deviation is 12/8 = 1.5.

The median absolute deviation is a different measure: find the median of the absolute distances from the dataset’s median. Because both the centre and the summary of distances use medians, it is generally less sensitive to extreme values than measures based on squared deviations. Some applications multiply it by a scaling factor, which must be stated.

The abbreviation MAD can refer to either mean absolute deviation or median absolute deviation in different contexts. Write the full name and define the calculation to avoid ambiguity. Both retain the original unit, but they are not interchangeable and need not produce the same numerical result.

How do you choose a spread measure and interpret rescaling?

Use range for a quick description of the observed extremes, IQR for the central spread when robustness is useful and standard deviation when variation around the mean is the focus. Reporting more than one can reveal whether a few extremes or the whole dataset drive the apparent variability.

Adding a constant to every observation shifts the distribution without changing its spread. Multiplying every value by a factor c multiplies range, IQR and standard deviation by |c|, while variance is multiplied by c². These relationships follow from how differences between values transform.

For example, converting lengths from metres to centimetres multiplies each observation by 100. The standard deviation also multiplies by 100, but the variance multiplies by 10,000. The physical variability has not changed; only the numerical unit scale has changed, which is why units must accompany reported dispersion values.

What mistakes should you avoid when measuring dispersion?

Sort the data before finding quartiles, and state the quartile convention when comparing results with another source or software. Different legitimate conventions can produce different small-sample quartiles. A discrepancy is not automatically an arithmetic error, but calculations within one analysis should use a consistent method.

Do not average signed deviations to measure spread, because deviations from the mean sum to zero. Use squares or absolute values to prevent cancellation. Also distinguish population and sample formulas, and avoid rounding the mean too early before calculating deviations, as that can distort the resulting variance.

Finally, interpret spread alongside centre, sample size and the shape of the distribution. A single measure cannot reveal clusters, gaps or every unusual observation. Numerical summaries are most informative when supported by a suitable display and a clear explanation of what was measured and how the summary was calculated.